Abstract
We consider the Cauchy problem for a hyperbolic equation with a small parameter multiplying the highest derivative. The inverse problem of finding an unknown function that is a coefficient of the equation and also occurs in the initial condition is posed. The values of the solution of the Cauchy problem and its derivative at x = 0 are given as additional information for solving the inverse problem. An iterative method for determining the unknown function is constructed, and its convergence is proved. Existence theorems are proved for the solution of the inverse problem.
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References
Lavrent’ev, M.M., Romanov, V.G., and Shishatskii, S.P., Nekorrektnye zadachi matematicheskoi fiziki i analiza (Ill-Posed Problems in Mathematical Physics and Calculus), Novosibirsk: Nauka, 1980.
Romanov, V.G., Obratnye zadachi matematicheskoi fiziki (Inverse Problems of Mathematical Physics), Moscow: Nauka, 1984.
Kabanikhin, S.I. and Lorenzi, A., Identification Problems of Wave Phenomena, Utrecht: VSP, 1999.
Belishev, M.I. and Gotlib, V.Yu., Dynamical variant of the BC-method: Theory and numerical testing, J. Inverse Ill-Posed Probl., 1999, vol. 7, no. 3, pp. 221–240.
Prilepko, A.I., Orlovsky, D.G., and Vasin, I.V., Methods for Solving Inverse Problems in Mathematical Physics, New York: Marcel Dekker, 2000.
Blagoveshchenskii, A.S., Inverse Problems of Wave Processes, Utrecht: VSP, 2001.
Isakov, V., Inverse Problems for Partial Differential Equations, New York: Springer, 2006.
Kabanikhin, S.I., Obratnye i nekorrektnye zadachi (Inverse and Ill-Posed Problems), Novosibisk: Sibirskoe Nauchn. Izd., 2008.
Denisov, A.M. and Solov’eva, S.I., Numerical determination of the initial condition in Cauchy problem for hyperbolic equation with a small parameter, Comput. Math. Model., 2018, vol. 29, no. 1, pp. 1–9.
Denisov, A.M. and Solov’eva, S.I., Numerical solution of inverse problems for a hyperbolic equation with a small parameter multiplying the highest derivative, Differ. Equations, 2018, vol. 54, no. 7, pp. 900–910.
Lattes, R. and Lions, J.-L., The Method of Quasi-Reversibility: Applications to Partial Differential Equations, Moscow: Mir, 1970.
Samarskii, A.A. and Vabishchevich, P.N., Chislennye metody resheniya zadach konvektsii-diffuzii (Numerical Methods for Solving Convection-Diffusion Problems), Moscow: Editorial URSS, 1999.
Korotkii, A.I., Tsepelev, I.A., and Ismail-zade, A.E., Numerical modeling of inverse retrospective heat convection problems with applications to geodynamic problems, Izv. Ural. Gos. Univ., 2008, no. 58, pp. 78–87.
Denisov, A.M., Asymptotic expansions of solutions to inverse problems for a hyperbolic equation with a small parameter multiplying the highest derivative, Comput. Math. Math. Phys., 2013, vol. 53, no. 5, pp. 580–587.
Belov, Yu.Ya. and Kopylova, V.G., Determination of the source function in a composite system of equations, Zh. Sib. Fed. Univ. Ser. Mat. Fiz., 2014, vol. 7, no. 3, pp. 275–288.
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Russian Text © The Author(s), 2019, published in Differentsial’nye Uravneniya, 2019, Vol. 55, No. 7, pp. 973–981.
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This work was supported in part by the Russian Foundation for Basic Research, project no. 17-01-00525.
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Denisov, A.M. Iterative Method for Solving an Inverse Problem for a Hyperbolic Equation with a Small Parameter Multiplying the Highest Derivative. Diff Equat 55, 940–948 (2019). https://doi.org/10.1134/S0012266119070073
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DOI: https://doi.org/10.1134/S0012266119070073